Generalizing Musical Intervals Dmitri Tymoczko Abstract Taking David Lewin’s work as a point of departure, this essay uses geometry to reexamine familiar music-theoretical assumptions about intervals and transformations. Section 1 introduces the problem of “trans- portability,” noting that it is sometimes impossible to say whether two different directions—located at two differ- ent points in a geometrical space—are “the same” or not. Relevant examples include the surface of the earth and the geometrical spaces representing n-note chords. Section 2 argues that we should not require that every interval be defined at every point in a space, since some musical spaces have natural boundaries. It also notes that there are spaces, including the familiar pitch-class circle, in which there are multiple paths between any two points. This leads to the suggestion that we might sometimes want to replace traditional pitch-class intervals with paths in pitch-class space, a more fine-grained alternative that specifies how one pitch class moves to another. Section 3 argues that group theory alone cannot represent the intuition that intervals have quantifiable sizes, proposing an extension to Lewin’s formalism that accomplishes this goal. Finally, Section 4 considers the analytical implications of the preceding points, paying particular attention to questions about voice leading. !"#$! %&'$(’) Generalized Musical Intervals and Transformations (henceforth GMIT ) begins by announcing what seems to be a broad project: modeling “directed measurement, distance, or motion” in an unspecified range of musical spaces. Lewin illustrates this general ambition with an equally general graph—the simple arrow or vector shown in Figure 1. This diagram, he writes, “shows two points s and t in a symbolic musical space. The arrow marked i sym- bolizes a characteristic directed measurement, distance, or motion from s to t. We intuit such situations in many musical spaces, and we are used to calling i ‘the interval from s to t’ when the symbolic points are pitches or pitch classes” (xi–xii).1 As is well known, Lewin goes on to model these general intuitions Thanks to Jordan Ellenberg, Noam Elkies, Tom Fiore, Ed Gollin, Rachel Hall, Henry Klumpenhouwer, Jon Kochavi, Shaugn O’Donnell, Steve Rings, Ramon Satyendra, Neil Weiner, and especially Jason Yust for helpful conversations. An earlier and more polemical version of this essay was delivered to the 2007 Society for Music Theory Mathematics of Music Analysis Interest Group. My thanks to those who encouraged me to write down my ideas in a less agonistic fashion. For more remarks on Lewin, see Tymoczko 2007, 2008a, and forthcoming. 1 The phrase “directed measurement, distance, or motion,” mathematically sophisticated readers, Dan Tudor Vuza (1988, or a close variant, appears at many points in GMIT (e.g., 278) and Julian Hook (2007a) take Lewinian intervals to 16, 17, 18, 22, 25, 26, 27). Elsewhere, Lewin describes represent quantified distances with specific sizes. See also intervals (group elements of a generalized interval system) Tymoczko 2008a. as representing “distances” (42, 44, 147). Among Lewin’s Journal of Music Theory 53:2, Fall 2009 DOI 10.1215/00222909-2010-003 © 2010 by Dmitri Tymoczko 227 228 J O URNAL of MUSI C THEO RY Figure 1. This graph appears as figure 0.1 in Lewin’s Generalized Musical Intervals and Transformations. It represents a directed motion from s to t in a symbolic musical space, with the arrow i representing “the interval” from s to t. using group theory, embodied in the twin constructions of the “generalized interval system” (GIS) and its slightly more flexible sibling, the “transforma- tional graph.”2 These innovations have proven to be enormously influential, forming the basis for countless subsequent music-theoretical studies.3 So per- vasive is Lewin’s approach that newcomers to music theory might well assume that group theory provides the broadest possible framework for investigating “directed measurement, distance, or motion.” It is interesting to note, however, that Lewin’s motivating question reca- pitulates one of the central projects of nineteenth-century mathematics: devel- oping and generalizing the notion of a vector. Intuitively, a vector is simply a quantity with both size and direction, represented graphically by an arrow such as that in Figure 1.4 Nineteenth-century mathematicians attempted to formalize this intuition in a variety of ways, including William Hamilton’s “quaternions” and Hermann Grassmann’s “theory of extended magnitudes,” now called the “Grassmann calculus.” The modern theory of vectors developed relatively late, primarily at the hands of Josiah Gibbs and Oliver Heaviside, and achieved wide- spread prominence only in the early decades of the twentieth century.5 While this development was occurring, other nineteenth-century mathematicians, such as Karl Friedrich Gauss, Bernhard Riemann, and Tullio Levi-Civita, were generalizing classical geometrical concepts—including “distance,” “straight line,” and “shortest path”—to curved spaces such as the surface of the earth. It was not until the twentieth century that the Gibbs-Heaviside notion of a “vector,” which requires a flat space, was defined as an intrinsic object in curved spaces. The result is a powerful set of tools for investigating Lewin’s question, albeit in a visuospatial rather than musical context. 2 Henry Klumpenhouwer (2006) asserts that Lewin’s book 4 “A vector, x, is first conceived as a directed line segment, opens with a “Cartesian” perspective that is eventually or a quantity with both a magnitude and direction” (Byron superseded by the later, “transformational” point of view. and Fuller 1992, 1). This interpretive issue is for the most part tangential to this 5 See Crowe 1994 for a discussion of quaternions, Grass- essay, since my points apply equally to Lewinian transforma- man calculus, and vector analysis. The conflict between tions (understood as semigroups of functions acting on a advocates of quaternions and vectors is a leitmotif in musical space). In Klumpenhouwer’s terms, I am suggesting Pynchon 2006. that even Lewinian transformations cannot capture certain elementary musical intuitions. 3 Indeed, one prominent theorist has declared that David Lewin “created the intellectual world I live in” (Rothstein 2003). Dmitri Tymoczko Generalizing Musical Intervals 229 The purpose of this essay is to put these two inquiries side by side: to con- trast Lewin’s “generalized musical intervals” with the generalized vectors of twentieth-century mathematics. My aims are threefold. First, I want to explore the fascinating intellectual-historical resonance between these two very differ- ent, and yet in some sense very similar intellectual projects—letting our under- standing of each deepen and enrich our understanding of the other. Second, I want to suggest that Lewin’s “generalized musical intervals” are in some ways less general than they might appear to be. For by modeling “directed motions” using functions defined over an entire space, Lewin imposes significant and nonobvious limits on the musical situations he can consider.6 Third, I want to suggest that there are circumstances in which we need to look beyond group theory and toward other areas of mathematics. Before proceeding, I should flag a few methodological and termino- logical issues with the potential to cause confusion. Throughout this essay I use the term “interval” broadly, to refer to vectorlike combinations of mag- nitude and direction: “two ascending semitones,” “two beats later,” “south by 150 miles,” “five degrees warmer,” “three days earlier,” and so on. (As this list suggests, intervals need not necessarily be musical.) I am suggesting that Lewin’s theoretical apparatus is not always sufficient for modeling “intervals” so construed. Some readers may object, preferring to limit the term “interval” to more specific musical or mathematical contexts. Since I have no wish to argue about terminology, I am perfectly happy to accede to this suggestion; such readers are therefore invited to substitute a more generic term, such as “directed distance,” for my term “interval” if they wish. Modulo this substitu- tion, my claim is that directed distances can be quite important in music- theoretical discourse and that geometrical ideas can help us to model them. I should also say that I am attempting to summarize some rather intri- cate mathematical ground. The mathematics relevant to this discussion is not that of elementary linear algebra but rather differential geometry, in which a vector space is attached to every point in a curved manifold. Comprehensibil- ity therefore requires that I elide some important distinctions, for instance, between torsion and curvature or between directional derivatives and finite motions along geodesics. I make no apologies for this: My goal here is to com- municate twentieth-century geometrical ideas to a reasonably broad music- theoretical audience, a process that necessitates a certain amount of simplifi- cation. I write for musical readers reasonably familiar with Lewin’s work but perhaps lacking acquaintance with Riemannian geometry, topology, or other branches of mathematics. Consequently, professional mathematicians may need to grit their teeth as I attempt to convey the spirit of contemporary geometry even while glossing over some of its subtleties. It goes without saying that I have done my best to avoid outright falsification. 6 In what follows, I use the term space broadly, to refer to a